Generally, the relevant credit history is the last 7 years. Resolving something that is 20 years old won't help. Incidentally, if the debt is really 20 years old, the company trying to collect the debt would probably keep all that you pay, which is why the company is willing to cut a deal.
If the debt is 20 years old it should not be reporting on your credit report. You should dispute the debt because Asset Acceptance may have "re-aged" the debt, claiming you made a recent payment. You can go to www.annualcreditreport.com to obtain a free copy of credit reports and also can do online disputes.
My comments are not legal advice and are for informational purposes only.